If a, b, and c are three different positive integers whose sum is prim

Question

If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime. 
(B) Each of a + 3, b + 3, and c + 3 is prime. 
(C) Each of a + b, a + c, and b + c is prime. 
(D) The average (arithmetic mean) of a, b, and c is prime. 
(E) a + b = c

From Manhattan's Challenge for this week.

manpreetsingh86 · Accepted Answer

we know that all prime numbers are odd except 2. if 2 is also a part of these 3 positive numbers, then sum will always be even. (even+ odd + odd= even) thus sum of a,b,c will be prime only if all 3 of them will be odd.

lets consider each option
a) 3,5,23 satisfy this condition. hence this option is possible.
b) as a,b,c are odd. therefore a+3, b+3, c+3 will always be even. hence this option is not possible
c) again as a,b,c are odd. therefore a+b,b+c,a+c will always be even. hence this option is not possible
d) sum of a,b,c is a prime number. therefore its sum will never be divisible by 3. (as prime numbers are divisible by 1 and itself). hence it will be a fraction. therefore this options is also not possible
e) sum of two odd numbers is always even. thus, this option is not possible at all.

xena123 · Answer

can you please explain as to why you have not taken 2?

MeRa2015 · Answer

I have tried to explain it using number properties -
3 different +ve integers (assume a, b, c) – sum is a prime number
All prime numbers except 2 are odd. 
Consider 2 -> There is no combination of a, b & c to get a sum of 2. (a, b and c have to be different and positive integers)
Now for a, b and c to add up to a prime number (>2) the possibilities are 
2 numbers are even positive integers and 1 is an odd positive integer ( E, E, O – i.e. assume a & b are even and c is odd)
OR
All 3 are odd positive integers (O, O, O – a, b & c are odd)

(A) E, E, O (even if 1 of the even numbers is 2 there is another number which cannot be prime) combination not true  but  O, O, O combination could be true.
(B) E+3 could be prime but O+3 is an even integer >2 so cannot be prime
So E,E,O combination not true (a+3 OK, b+3 OK but c+3 – cannot be prime ) and O,O,O (a/b/c+3 cannot be prime)combination is also not true.
(C) E+O could be prime but O+O cannot be 
So E,E,O (a+b cannot be prime) combination and O,O,O (a+b, b+c and c+a cannot be prime) combination not true
(D) If the average of 3 numbers is a prime number then the sum of the 3 numbers is a multiple of 3 and the arithmetic mean. So the sum cannot be even – therefore this option is also not possible.
(E) If a+b=c then a+b+c = 2c which is an even no and hence this option is not possible
Answer (A) is the only choice which could be true.

PareshGmat · Answer

Lets say a = 11, b = 13, c = 17

(A) Each of a, b, and c is prime >>>>>>>>>>>>>>>>>>>>>>>>>>. YES  
(B) Each of a + 3, b + 3, and c + 3 is prime. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> No in this case
(C) Each of a + b, a + c, and b + c is prime. >>>>>>>>>>>>>>>>>>>>>>>>>>>>> Not necessary
(D) The average (arithmetic mean) of a, b, and c is prime. >>>>>>>>>>>>>>>>>>>> Its fraction in this case
(E) a + b = c >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Not required

Answer = A

mvictor · Answer

the question asks could be!!!!
this means that if at least one option works, it is the answer.
3, 7, 19 satisfies the condition, since 29 is a prime number. Since the question asks for a could be - we know automatically that A is the answer.

SOUMYAJIT_ · Answer

I understood the question wrong now that you explain what "could" means. I got the meaning. Any other such things that I should remember?

bkpolymers1617 · Answer

If a, b, and c are three different positive integers whose sum is prime, which of the following statements could be true?

(A) Each of a, b, and c is prime.

(B) Each of a + 3, b + 3, and c + 3 is prime.

(C) Each of a + b, a + c, and b + c is prime.

(D) The average (arithmetic mean) of a, b, and c is prime.

(E) a + b = c

pushpitkc · Answer

Since a,b,c are 3 different postitive integers, and its sum is prime.
In Option A, consider a=3,b=5,c=11, sum of a,b and c is 19(which is prime)

Hence Option A(Each of a,b and c are prime) is possible.

sashiim20 · Answer

Let  a = 3 ,  b = 5  and  c = 11

a + b + c = 3 + 5 + 11 = 19  ---------- (Prime number)

Lets check the options.

(A) Each of a, b, and c is prime.

Let  a = 3 ,  b = 5  and  c = 11  ---------- (a,b and c is prime)--------- (Could be true)

(B) Each of a + 3, b + 3, and c + 3 is prime.

From (i) a + 3, b + 3 and c + 3 gives ( 6, 8 and 14) respectively ----------- (Not prime)

(C) Each of a + b, a + c, and b + c is prime.

From (i)  a + b = 8

a + c = 14

b + c = 16  ------- (Not prime)

(D) The average (arithmetic mean) of a, b, and c is prime.

From (i)  \frac{3 + 5 + 11}{3} = \frac{19}{3}  ------------- (Not prime)

(E)  a + b = c  ==> From (i)  3 + 5 = 8  not 11 ------- ( Not true)

Therefore Only A could be true . Answer (A)...

Bunuel · Answer

Merging topics. You are permanently violating our posting rules (https://gmatclub.com/forum/rules-for-po ... 33935.html). This post for example, was posted in DS instead of PS and is also a duplicate. Note that ignoring or breaking these rules will likely lead to removed posts, warnings, and eventual bans.

EMPOWERgmatRichC · Answer

Hi All,

This question has some interesting "restrictions" to it, but it's perfect for TESTing VALUES (even though you'll have to do a bit more work than normal to find a TEST that "fits").

We're told that A, B and C are 3 DIFFERENT positive integers whose SUM is PRIME.

Since the sum has to be prime, we should be looking for an ODD total. Since the answer choices emphasize the idea of prime numbers, you should think about "brute force-ing" a series of 3 odd numbers (likely primes) to see if you can get a match.

Without too much trouble, I got to 3, 5 and 11….

3+5+11 = 19 = a sum that is prime

Using these values, only one answer is true....

Final Answer:  A

GMAT assassins aren't born, they're made, 
Rich

KarishmaB · Answer

Number plugging could be tedious in this question since we are asked for "could be true" i.e. there could be some values for which this relation could hold. We will prove that 4 options are not possible. Automatically the fifth option will be possible.

a, b, and c are three different positive integers whose sum is prime. So e.g. 1, 2, 4 add up to give 7.

We know that only 2 is an even prime number. Rest all prime numbers are odd. So sum will be odd.

There are 2 ways to obtain an odd sum:
 Case 1: Odd + Odd + Odd = Odd
Case 2: Odd + Even + Even = Odd

Hence either all a, b, and c are odd, OR 1 of them is odd and other 2 are even.

(B) Each of a + 3, b + 3, and c + 3 is prime.  
At least one number will be odd (in both cases above) which when added to 3 will give an even sum (hence not a prime number).
This is not possible.

(C) Each of a + b, a + c, and b + c is prime.  
Again, Odd + Odd = Even (Case 1) and Even + Even = Even (Case 2) so in either case, one of these sums will not be prime.
This is not possible.

(D) The average (arithmetic mean) of a, b, and c is prime.  
The sum of a, b, c is prime so it will not be divisible by 3 at all. Hence it won't even be an integer. 
This is not possible.

(E) a + b = c  
In case 1, Odd + Odd = Even, hence this is not possible.
In case 2, Odd + Even = Odd (not possible) and Even + Even = Even (Not possible)
This is not possible.

Answer must be (A).

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If a, b, and c are three different positive integers whose sum is prim

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If a, b, and c are three different positive integers whose sum is prim

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